Universal property


by Patrick Stevens Aug 26 2016 updated Dec 31 2016

A universal property is a way of defining an object based purely on how it interacts with other objects, rather than by any internal property of the object itself.

[summary: In Category theory, we attempt to avoid thinking about what an object is, and look only at how it interacts with other objects. A universal property is a way of defining an object not in terms of its "internal" properties, but instead by its interactions with the "universe" of other objects in existence.]

In Category theory, we attempt to avoid thinking about what an object is, and look only at how it interacts with other objects. It turns out that even if we're not allowed to talk about the "internal" structure of an object, we can still pin down some objects just by talking about their interactions. For example, if we are not allowed to define the Empty set as "the set with no elements", we can still define it by means of a "universal property", talking instead about the functions from the empty set rather than about the elements of the empty set.

This page is not designed to teach you any particular universal properties, but rather to convey a sense of what the idea of "universal property" is about. You are supposed to let it wash over you gently, without worrying particularly if you don't understand words or even entire sentences.


The above examples show that the ideas of category theory are very general. For instance, the third example captures the idea of a "free" object, which turns up all over Abstract algebra.

Definition "up to isomorphism"

[todo: explain that we only usually get things defined up to isomorphism, and what that means anyway]

Universal properties might not define objects

Universal properties are often good ways to define things, but just like with any definition, we always need to check in each individual case that we've actually defined something coherent. There is no silver bullet for this: universal properties don't just magically work all the time.

For example, consider a very similar universal property to that of the Empty set (detailed here), but instead of working with sets, we'll work with fields, and instead of functions between sets, we'll work with field homomorphisms.

The corresponding universal property will turn out not to be coherent:

The initial field %%note: Analogously with the empty set, but fields can't be empty so we'll call it "initial" for reasons which aren't important right now.%% is the unique field $~$F$~$ such that for every field $~$A$~$, there is a unique field homomorphism from $~$F$~$ to $~$A$~$.

%%%hidden(Proof that there is no initial field): (The slick way to communicate this proof to a practised mathematician is "there are no field homomorphisms between fields of different [field_characteristic characteristic]".)

It will turn out that all we need is that there are two fields [4zq $~$\mathbb{Q}$~$] and $~$F_2$~$ the field on two elements. %%note: $~$F_2$~$ has elements $~$0$~$ and $~$1$~$, and the relation $~$1 + 1 = 0$~$.%%

Suppose we had an initial field $~$F$~$ with multiplicative identity element $~$1_F$~$; then there would have to be a field homomorphism $~$f$~$ from $~$F$~$ to $~$F_2$~$. Remember, $~$f$~$ can be viewed as (among other things) a Group homomorphism from the multiplicative group $~$F^*$~$ %%note: That is, the group whose Underlying set is $~$F$~$ without $~$0$~$, with the group operation being "multiplication in $~$F$~$".%% to $~$F_2^*$~$.

Now $~$f(1_F) = 1_{F_2}$~$ because the image of the identity is the identity, and so $~$f(1_F + 1_F) = 1_{F_2} + 1_{F_2} = 0_{F_2}$~$.

But field homomorphisms are either injective or map everything to $~$0$~$ (proof); and we've already seen that $~$f(1_F)$~$ is not $~$0_{F_2}$~$. So $~$f$~$ must be injective; and hence $~$1_F + 1_F$~$ must be $~$0_F$~$ because $~$f(1_F + 1_F) = 0_{F_2} = f(0_F)$~$.

Now examine $~$\mathbb{Q}$~$. There is a field homomorphism $~$g$~$ from $~$F$~$ to $~$\mathbb{Q}$~$. We have $~$g(1_F + 1_F) = g(1_F) + g(1_F) = 1 + 1 = 2$~$; but also $~$g(1_F + 1_F) = g(0_F) = 0$~$. This is a contradiction. %%%